W2_Equations and Inequalities

Session Information

  • Title: Fundamentals of Mathematics 1 Session 2: Equations and Inequalities

  • Instructor: Erwan Lamy

  • Institution: ESCP Business School

  • Locations: Berlin, London, Madrid, Paris, Turin, Warsaw

Objectives

  • To solve:

    • Linear equations

    • Quadratic equations

  • To model situations described by linear or quadratic equations

  • To solve linear inequalities in one variable and introduce interval notation

  • To model real-life situations in terms of inequalities

  • To solve equations and inequalities involving absolute values

Outline

  1. Equations, in Particular Linear Equations (chp 0.7)

  2. Radical and Quadratic Equations (chp 0.8)

  3. Applications of Equations (chp 1.1)

  4. Linear Inequalities (chp 1.2)

  5. Applications of Inequalities (chp 1.3)

  6. Absolute Value (chp 1.4)

Equations and Linear Equations

Definition of an Equation

  • An equation is a statement that two expressions are equal, e.g.,

    • Example: −1 = 9

  • Sides are two expressions separated by the equality sign ( = ).

  • Contains at least one variable (x, y, z, ...).

    • Example: In x + 2 = 3, x is the variable; 2 and 3 are constants.

Solving an Equation

  • To solve means to find all values of its variables that satisfy the equation, called solutions.

  • The solution set contains all possible solutions.

  • Example: x = 1 is a solution for x + 3 = 4.

Operations that Guarantee Equivalence

  • Operations to maintain equivalence:

    • Adding/subtracting the same polynomial to/from both sides.

    • Multiplying/dividing both sides by the same nonzero constant.

    • Replacing either side with an equal expression.

  • Examples confirming these operations maintain equation equivalence.

Operations that May Not Produce Equivalent Equations

  1. Multiplying/dividing by an expression involving the variable

  2. Raising both sides to equal powers

  • Example: Different operations may lead to non-equivalent equations.

Linear Equations

  • A linear equation in variable x can be expressed as ax + b = 0 with a and b as constants. This is known as a first-degree equation.

Applications of Equations

  • Modeling involves translating verbal problems into mathematical equations.

  1. Identify unknowns and relevant variables.

  2. Write relationships among variables.

  3. Solve equations.

  4. Validate solutions for realism.

Applications of Linear Inequalities (chp 1.3)

  • Define an inequality, proven by examples like:

    • If a < b, then a + c < b + c for any c.

Examples of Problems

  1. Profit Problem: Formulating an equation to determine the number of units for profit.

  2. Investment Problem: Using equations to find investment allocations in multiple ventures.

  3. Real Estate Problem: Utilizing linear inequalities to maximize profit from renting apartment units.

Absolute Value (Chapter 1.4)

  • Defined as the distance of x from 0 on the number line:

    • |x| = { x if x ≥ 0; -x if x < 0 }

  • Absolute value equations demonstrate that certain equations yield no solution if applying absolute value directly is not valid.

  • Inequalities involving absolute values can be solved via interval notation.